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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs
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by Richard W. Kenyon and David B. Wilson PDF
J. Amer. Math. Soc. 28 (2015), 985-1030

Abstract:

We show how to compute the probabilities of various connection topologies for uniformly random spanning trees on graphs embedded in surfaces. As an application, we show how to compute the “intensity” of the loop-erased random walk in $\mathbb {Z}^2$, that is, the probability that the walk from $(0,0)$ to $\infty$ passes through a given vertex or edge. For example, the probability that it passes through $(1,0)$ is $5/16$; this confirms a conjecture from 1994 about the stationary sandpile density on $\mathbb {Z}^2$. We do the analogous computation for the triangular lattice, honeycomb lattice, and $\mathbb {Z}\times \mathbb {R}$, for which the probabilities are $5/18$, $13/36$, and $1/4-1/\pi ^2$ respectively.
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Additional Information
  • Richard W. Kenyon
  • Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
  • MR Author ID: 307971
  • David B. Wilson
  • Affiliation: Microsoft Research, Redmond, Washington 98052
  • Received by editor(s): July 19, 2011
  • Received by editor(s) in revised form: August 12, 2013, February 12, 2014, May 6, 2014, May 22, 2014, and June 5, 2014
  • Published electronically: October 21, 2014
  • Additional Notes: The research of the first author was supported by the NSF
  • © Copyright 2014 by the authors. This paper or any part thereof may be reproduced for non-commercial purposes.
  • Journal: J. Amer. Math. Soc. 28 (2015), 985-1030
  • MSC (2010): Primary 60C05, 82B20, 05C05, 05C50
  • DOI: https://doi.org/10.1090/S0894-0347-2014-00819-5
  • MathSciNet review: 3369907